Two Red Lines in a Rectangle

Imagine a rectangle with sides \(\overline{\text{AB}}\), \(\overline{\text{BC}}\), \(\overline{\text{CD}}\), and \(\overline{\text{DA}}\). Side \(\overline{\text{AB}}\) has a length of \(4\) while \(\overline{\text{BC}}\) has a length of \(6\).

Put point E\text{E} on AD\overline{\text{AD}}. Draw a line from E\text{E} to B\text{B} and draw another line from E\text{E} to C\text{C}.

If BEC\angle\text{BEC} has a measurement of 65°65°, what is the length of line segments EB\overline{\text{EB}} and EC\overline{\text{EC}}?

Rectangle \(\text{ABCD}\) Rectangle ABCD\text{ABCD}

How can I solve for the lengths of the red lines? A step-by-step solution will be very much appreciated.

Here's the link of an image of described shape:

Note by Kaizen Cyrus
1 year ago

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Here's a hint: let DE\overline{DE} be xx. Then, AE=6x\overline{AE} = 6-x. We can use the Pythagorean Theorem to express EB\overline{EB} and EC\overline{EC} in terms of xx. Then, you can use Cosine Law on EBC\triangle EBC and solve for xx. Plug the result back into EB\overline{EB} and EC\overline{EC}, and the problem is solved!

Elijah L - 1 year ago

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AE\overline{AE} is equal to 6x6-x? But AD\overline{AD} is 66 and you said let it be xx.

Kaizen Cyrus - 1 year ago

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Whoops, my mistake. I meant let DE\overline{DE} be xx. Everything else stays the same.

Elijah L - 1 year ago

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